The effect of Proportional Editing in Random mode is not as strong as I expected. How can I make it more intense?
After having initiated the G ("move") action, scroll up the mouse wheel ( wheel) to enlarge the area of influence of the proportional edit.

While scrolling (up or down) you'll see a circle representing roughly to what spatial extent the other vertices will be affected. Make sure it is large enough to encompass several of the other vertices.

Then just keep moving the selected vertex to see the effect. (As Ashutosh Budhdeo noted, you may want to use Z during the "move" action to constrain the movement to the vertical axis).